Einsteen, I'm glad you're thinking about this, and I applaud you for struggling through language difficulties, but you're just not thinking about it clearly. Let me try to help.
Upper block falls
This is assumed by TV coverage, it does not explain why this happened, ok but you have to start with something, allright then.
Where else would the block go? I'm confused. Gravity will pull it downward.
Non-elastic collision
This is not what we see from the same TV coverage, as an object breaking floors below, all the dust and crushed concrete staying 'under' the block and traveling in the same direction. If you do a gedankenexperiment in which you assume this then it's ok, but for a first approximation of the theoretical minimum collapse time you already should take other factors into account, the time could be extremely dependent on these factors or maybe not.
No, einsteen, you're wrong. The collision is inelastic. The collision can only be
elastic, the alternative, if no energy is dissipated in breaking or bending things when they hit. This is impossible. Greening's asumption is correct. His mathematics allow for fragmentation.
Tacitly assumed that the impulsive delivered by the impact is sufficient to rupture...
Why?
Because in this part of the paper he's considering timing. He addresses the energy required for progressive collapse elsewhere. Thoroughly.
[The second stage of collapse
Absolutely relevant for the seismic data, but not really relevant for the speed at which the buildings come down.
The timing information is derived from seismic data. It's relevant to both.
For now it is sufficient to note that the collapse times calculated without allowing for E1 are already in reasonable agreement with the observed collapse times.
Of course because this is the theoretical minimum time and that should be near free-fall time because in fact section 3.0 of the paper assumes no resistance in any way, in fact point masses with non-elastic collapses merging into each other without any of the mass moving at an other directory
Wrong. The paper assumes resistance from structure and inertia of lower floors. It does not take wind resistance into account -- which is correct. The floors are not falling into a free-stream of air, they're falling into structure, and the structure's resistance is accounted for.
But the initial kinetic energy Ti is equal to (1/2)Nm_f u^2 so the fractional conversion, fc, of kinetic energy to heat is simply,
fc = Q/Ti = 1/(1 + N)
This is an amazing result IMO, but indeed under the assumption no mass is scattered away from floor N to floor N-1 to ... to floor 1
If you notice, the energy budget is far higher than needed. You will see a little bit of mass shed off the sides, but not much. Inelastic, remember? Stuff doesn't bounce off at full speed. You only lose a little mass on the edges, or the unlucky piece of steel that breaks and throws fragments.
You could easily estimate how much fell off the sides by comparing the distribution of debris after the collapse was over. Nearly all of it was in one central pile. The losses were small, perhaps 10% in mass.
If we assume 50 % of this energy was available to crush concrete, we have 1.2 x 10^9 J available for WTC 1, and 2.5 x 10^9 J available for WTC 2. This is sufficient to crush the concrete on the impacted floor to 175 micrometer particles. Consider now the newly formed mass of (14 + 1) floors of WTC 1, and (29 + 1) floors of WTC 2, impacting on the floor below.
An amazing amount of energy indeed. Again all crushed concrete is assumed to form a new mass to crush the next floor. Check some videos and detailed pictures, it's absolutely impossible to consider this as a first approximation, the mass scatters in all directions, even upwards. Of course the conservation of mass can't be violated, but the part traveling down together with the initial upperblock is not
(n+1)m_floor
but should be
(wn+1)m_floor,
with w the fraction of the mass traveling downwards. Greening didn't use that, also note that w is not constant, it will be a complicated function (dependent on some variables) for which we only know 0< w<1, it should be estimated.
The bulk of the tower's mass is not concrete, and is not crushable. Your fudge factor W will be close to 1.
Also remember that mass that "scatters" will only do so by impacting the lower structure very hard. It has already contributed most of the energy that a piece that stayed with the column would have. Your losses are just not high enough to invalidate his hypothesis.
Further small particles are very sensitive to air resistance and the rest of the crap in the air and the so-called terminal speed is reached very quickly, I'm not sure if the factor w can take that into acount or if another factor is needed for that somewhere. Maybe I'm wrong also now and all these factors seem to have not much influence on the theoretical minimum, but it was just something I thought after reading a little bit.
Again, only the stuff that falls off the sides is in a free-stream of air. And we've already discounted it. Fine pieces still in the core structure do not experience meaningful drag.
If you want to explore the effect of pieces falling off the sides, just re-run Greening's calculation but reduce the mass of each floor, say by 5%, 10%, on up to 100%. Find how the calculation behaves as a function of debris loss.
Then take a look at the collapse pile, estimate a debris loss fraction, and see where you wind up with his collapse equations.
Having done this, I believe you'll find that his equations are quite credible, especially considering they are a simplification.
I don't like this way of conversation. Why that aggresive, what did I do to you ? I know enough of Newtonian mechanics.
